Trying to get to the bottom of 'what, ultimately, are zeta functions?'. Could all partition functions used in statistical mechanics, etc. be considered zeta functions in some appropriate setting?
 

I seem to remember that there was some unresolved difficulty with Spector's work. Go back to that.
 

Thinking about periodic orbits as fixed points in some sort of 'loop space'. Is this standard? Is it appropriate to relate energy levels to 'stable states' which are fixed points in some sort of state space? If so, could both sides of trace formulae be regarded as sums over fixed points?
 

It's about time you had a page relating modular forms, theta functions, the functional equation, etc. There are those highly relevant Dutch transparencies, also notes from Ninham, remarks on fnleqn.htm, remarks by Julia. This is getting into the sort of thing Ramanujan was working on. Somehow relevant to string theory - Veneziano amplitudes or something. A lot to be done here.